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I did the following exercise, but I really don't know if it's okay

A mouse enters a box with 9 gaps as follows:

enter image description here

The probability to go from one state to another is equally probable depending on the space in which it is. Additionally, in space 9 there is a cat that keeps in that space and in space 1 there is cheese.

The question is: What is the probability that the mouse finds the cheese before the mouse eats it?

My attempt:

Construct the matrix of transition probabilities as follows:

\begin{equation*} \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/3 & 0 & 1/3 & 0 & 1/3 & 0 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0\\ 1/3 & 0 & 0 & 0 & 1/3 & 0 & 1/3 &0 & 0\\ 0 & 1/4 & 0 & 1/4 & 0 & 1/4 & 0 & 1/4 & 0\\ 0 & 0 & 1/3 & 0 & 1/3 & 0 & 0 & 0 & 1/3\\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 1/2 & 0\\ 0 & 0 & 0 & 0 & 1/3 & 0 & 1/3 &0 & 1/3\\ 0 & 0 & 0 & 0 & 0 & 0 & &0 & 1\\ \end{pmatrix} \end{equation*}

Then I built the canonical matrix by changing columns 1 and 8 and rows 1 and 8. Then calculate the fundamental matrix: $N = (I-Q)^{-1}$, which is:

\begin{equation*} \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1/3 & 0 & 1/3 & 0 & 1/3 & 0 & 0\\ 0 & 1/2 & 0 & 0 & 0 & 1/2 & 0 \\ 1/3 & 0 & 0 & 0 & 1/3 & 0 & 1/3 \\ 0 & 1/4 & 0 & 1/4 & 0 & 1/4 & 0 \\ 0 & 0 & 1/3 & 0 & 1/3 & 0 & 0 \\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0\\ \end{pmatrix} \end{equation*}

And multiplying it by vector 1,

\begin{equation*} \begin{pmatrix} 29/6 \\ 49/10\\ 59/10\\ 137/30\\ 29/5\\ 49/10\\ 57/10\\ \end{pmatrix} \end{equation*}

Therefore, the answer is: The probability that the mouse finds the cheese (it is in cell 1) before it finds the cat (it is in cell 9) is $ 29 / 6$

Katherine
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    Your first concern should be that $29/6$ is not a probability. Probabilities are always between $0$ and $1$, with $1$ corresponding to something that always happens. $29/6$ is approximately $5.833$. – Misha Lavrov Apr 18 '21 at 20:39
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    Where does the mouse start? Your box doesn't seem to have an entrance. – Troposphere Apr 18 '21 at 20:42

1 Answers1

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The probability depends on where the mouse enters the box:

  • In space 1, the probability that the mouse finds the cheese before being eaten by the cat is $1$.
  • In space 9, that probability is $0$.
  • In spaces 3, 5, and 7, that probability is $\frac12$, by symmetry: the cat and the cheese are symmetric about that diagonal, and going in the cat's direction is just as likely as going in the cheese's direction.
  • In spaces 2, 4, 6, and 8, you can find the probability by looking at where the mouse will go next, and using the probabilities in those spaces (which are all known).

Finally, after finding the probability that the mouse finds the cheese before being eaten by the cat, we realize that this wasn't the question asked. The answer to the question

What is the probability that the mouse finds the cheese before the mouse eats it?

is $1$: the mouse can't very well eat the cheese before finding it.

Misha Lavrov
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    Also helps to remark that the answer from $2$ is the same as the answer from $4$, and the answer from $6$ is the same as that from $8$ (and that, if $\rho$ is the answer from $2$, the answer from $6$ is $1-\rho$). – lulu Apr 18 '21 at 20:50